## Search found 6 matches

- Wed Dec 23, 2020 9:36 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2015#6
- Replies:
**4** - Views:
**4023**

### Re: BdMO National Secondary 2015#6

Thank you for the suggestion. I will do that.

- Tue Dec 22, 2020 10:08 pm
- Forum: Combinatorics
- Topic: A combi salad from mathematical olympiad treasures
- Replies:
**2** - Views:
**3574**

### A combi salad from mathematical olympiad treasures

Let n and k be two natural numbers and let S be a set of n points such that

(a) no three points of S are collinear.

(b) for any point P of S there are at least k points of S which are equidistant from P.

Prove that k<1/2+(2n)^(1/2)

(a) no three points of S are collinear.

(b) for any point P of S there are at least k points of S which are equidistant from P.

Prove that k<1/2+(2n)^(1/2)

- Mon Dec 21, 2020 7:39 pm
- Forum: Number Theory
- Topic: A problem from USAMO 2003
- Replies:
**5** - Views:
**3767**

### A problem from USAMO 2003

Prove that for all positive integers n there is an n-digit multiple of 5^n all the the digits of which is odd.

- Mon Dec 21, 2020 12:36 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2020 P7
- Replies:
**2** - Views:
**2692**

### Re: BdMO National Secondary 2020 P7

Thanks to Anindya Biswas for solving a general form of the problem

- Sat Dec 19, 2020 9:33 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2015#6
- Replies:
**4** - Views:
**4023**

### Re: BdMO National Secondary 2015#6

Let X be the intersection of AD and BC. Then XA/XB=XD/XC =(XA-XD)/(XB-XC)=AD/BC=70/50=7/5. Then P lies on the bisector of <AXB. Therefore P is the intersection of side AB and the bisector of its opposite angle. So AP/PB=XA/XB=7/5. So AP=7AB/(7+5)=7×92/12=161/3.

So the answer is 161+3=164

So the answer is 161+3=164

- Sun Dec 13, 2020 8:08 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National Secondary 2020 P10
- Replies:
**5** - Views:
**5770**

### Re: BdMO National Secondary 2020 P10

Let A be the set of all such paths and let B be the set of all sequence of coordinates (x,y) such that both x and y are primes , if (x1,y1) is an entry of the sequence then then the next entry is (x2,y2) where x1=x2 and y2 is the smallest prime greater than y2 or vice versa and the first entry of th...